the chemical rate constants that relate these transport coefficients to the .... Since TrËÏ(0) ËR(t) = 0, the average of Eq. (21) over the initial nonequilibrium ...
Chemical Rate Laws and Rate Constants Raymond Kapral, Styliani Consta and Liam McWhirter Chemical Physics Theory Group, Department of Chemistry University of Toronto, Toronto M5S 1A1, Canada
1. – Introduction Chemical reactions are described by mass action kinetic equations that specify how the mean concentrations of chemical species vary with time. Under what circumstances is such a description possible? How may one compute the values of the rate constants that enter these equations from a knowledge of the microscopic properties of the system? Complete answers to these questions cannot be given. However, for systems close to equilibrium where linearized versions of mass action laws apply, one may derive generalized forms of the rate laws from the microscopic evolution equations. The investigation of the conditions under which these generalized laws reduce to mass action kinetics supplies the answer to the first question. In the course of this derivation one obtains autocorrelation function expressions for the chemical rate constants that relate these transport coefficients to the microscopic dynamics. These expressions, while exact, are formidable to compute for a many-body quantum system. Nevertheless, these correlation expressions form the starting point for a discussion of various approximate schemes for the computation of rate constants. The derivation of the generalized chemical rate law and a discussion of the conditions under which reduction to the phenomenological form is possible are presented in Sec. 2 for a general quantum mechanical system. A detailed discussion of the properties of the rate kernel that enters this description is also presented in this section. Section 3 specializes these results to classical systems and the results are illustrated with a model of diffusive barrier crossing and a molecular dynamics study of ion solvation dynamics in water clusters. The next two sections deal with mixed quantum-classical systems. Section 4 considers the simpler case of adiabatic dynamics. Following a discussion of the reduction of the full quantum dynamics to the adiabatic mixed quantum-classical limit, a description of proton transfer in a cluster composed of polar solvent molecules is given as an example of an application. The last section considers nonadiabatic dynamics in the mixed quantum-classical limit. A discussion of some of the approximations that lead to surface hopping methods are presented and the cluster proton transfer problem is revisited to show the effects of nonadiabaticity on this reaction. c Societ`
a Italiana di Fisica
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RAYMOND KAPRAL, STYLIANI CONSTA AND LIAM MCWHIRTER
2. – Derivation of Chemical Rate Laws . 2 1. Phenomenological description. – Consider a reaction among n chemical species Xi , i = 1, . . . , n of the form (1)
kf
ν 1 X1 + ν 2 X2 + · · · ν n Xn * ) ν¯1 X1 + ν¯2 X2 + · · · ν¯n Xn , kr
characterized by the forward and reverse rate constants kf and kr . The stoichiometric coefficients are νi and ν¯i for reactants and products, respectively. The mass action rate ¯i of the chemical species is law for the average numbers N n n Y Y ¯i (t) dN ¯ ν¯j (t) = (νi − ν¯i )J , ¯ νj (t) + kr N (2) N = (νi − ν¯i ) −kf j j dt j=1 j=1 The second equality defines the reaction rate J which is independent of the species label in view of the constraints on the particle number changes implied by Eq. (1). For a reacting system at constant temperature T and volume V , the entropy change as a result of reactions is
(3) where the chemical affinity, (4)
T
n X ¯i ds dN =− µi = JA , dt dt i=1
A=
n X i=1
µi (νi − ν¯i ) ,
is the thermodynamic driving force of the chemical reaction, which vanishes at equilibrium. Equation (2) can be used to define the progress variable χ(t) ¯ which characterizes the extent of reaction [1]: (5)
¯i (t) dχ(t) ¯ dN = (νi − ν¯i )−1 =J . dt dt
This equation may be integrated from time t to t = ∞, where the system is in equilibrium and χ(∞) ¯ = 0, to obtain (6)
¯i (t) − N ¯ eq ) . χ(t) ¯ = (νi − ν¯i )−1 (N i
Only a single dynamical variable χ ¯ is needed to characterize the extent of reaction. We shall be concerned with reactive systems that are perturbed slightly from chemical equilibrium so that the chemical affinity is small. We also assume that the reactive species are dilute in some chemically inert solvent so that the chemical potentials take ¯i . (1 ) Expanding the average particle numbers in the simple form µi = µ0i + kT ln N terms of the progress variable and linearizing in χ, ¯ we obtain (7)
dχ(t) ¯ = −k χ(t) ¯ , dt
(1 ) For nonideal systems the chemical potential is expressed in terms of the activity. One may also question whether the phenomenological rate law, Eq. (2), should also be written in terms of activities. For a discussion of this issue see Ref. [2].
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CHEMICAL RATE LAWS AND RATE CONSTANTS
where the rate constant k is given by " # n n n Y X Y ¯j eq νi νj eq ν¯i ν ¯ ¯ (Ni ) ¯ eq − kr k= (νj − ν¯j ) kf (8) (Ni ) ¯ eq . Nj Nj i=1 j=1 i=1 From this expression we see that the rate constant k depends on both the forward and reverse rate constants and, in general, on the equilibrium concentrations. In the simple case of a linear interconversion reaction of the form kf
A* )B,
(9)
kr
which shall primarily concern us here, k takes the simple form k = kf + kr . Finally, we note that Eq. (7) may be integrated to yield Z t (10) χ(t) ¯ − χ(0) ¯ = −k dt0 χ(t ¯ − t0 ) , 0
which will prove to be a useful form in the discussion that follows. . 2 2. Nonequilibrium initial ensemble. – From Eqs. (3) and (5) we observe that the affinity A, the thermodynamic driving force, is conjugate to the flux of the progress variable χ. ¯˙ In order to appreciate some of the ingredients necessary to construct a microscopic description of the reactive process we consider an initial nonequilibrium ensemble where only the progress variable is constrained to deviate from its zero equilibrium value. We consider a quantum mechanical system and suppose that the microscopic Hermitian operator corresponding to the progress variable is χ. ˆ (Henceforth, a hat on a symbol will denote a quantum mechanical operator.) We shall discuss specific forms for χ ˆ later. ( 2 ) The nonequilibrium ensemble for the system at constant temperature T and volume V may be constructed from the usual canonical distribution by appending an additional ˆ of the system: term χA ˆ to the Hamiltonian H ˆ
ˆ e−β(H−χA)
ρˆ(0) =
(11)
ˆ χA) ˆ Tre−β(H−
,
where β = (kT )−1 as usual. It then follows that the average value of the progress variable is given by (12)
χ(t) ¯ = Trχ(t)ˆ ˆ ρ(0) .
Using the operator identity (13)
ˆ
ˆ
ˆ e−β(H−χA) = e−β H +
Z
β
ˆ
ˆ
ˆ dλe−λ(H−χA) χAe ˆ −(β−λ)H ,
0
we may write the linearized form of ρˆ(0) as (14)
ρˆ(0) = ρˆe +
Z
β 0
ˆ
ˆ
dλe−λH χe ˆ λH ρˆe A ,
(2 ) A formal discussion of species operators may be found in Ref. [3]
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RAYMOND KAPRAL, STYLIANI CONSTA AND LIAM MCWHIRTER
from which it follows that (15)
χ(t) ¯ = Tr
Z
β 0
ˆ
ˆ
−λH dλχ(t)e ˆ χe ˆ λH ρˆe A = (χ(t), ˆ χ)βA ˆ .
Here ρˆe is the equilibrium density matrix, ˆ
ρˆe =
(16)
e−β H Tre−β Hˆ
,
and the Kubo transformed correlation function is defined as [4] (17)
ˆ B ˆ † ) = β −1 (A,
Z
β
ˆ
ˆ
ˆ −λH B ˆ † eλH ρˆe . dλAe
0
From Eq. (15) one observes that near equilibrium the nonequilibrium average of χ(t) ˆ is given by the autocorrelation function describing the fluctuations of the progress variable about equilibrium. . 2 3. Rate law derivation. – To derive the chemical rate law [5] one starts from the Heisenberg equation of motion for χ, ˆ (18)
dχ(t) ˆ i ˆ ˆ = [H, χ(t)] ˆ = iLˆχ(t) ˆ = eiLt iLˆχ ˆ, dt h ¯
and extracts the evolution proportional to χ(t) ˆ using projection operator methods. [6, 7] In view of the above considerations an appropriate projection operator is (19)
ˆ = (O, ˆ χ)( Pˆ O ˆ χ, ˆ χ) ˆ −1 χ ˆ,
ˆ onto χ. since Pˆ just projects O ˆ Substituting the operator identity Z t ˆ ˆ ˆ ˆˆ ˆ Le ˆ iQˆ Lτ eiLt = dτ eiL(t−τ ) Pi (20) + eiQLt , 0
ˆ = 1 − P, ˆ into the last equality in Eq. (18) we obtain the generalized Langevin where Q equation for χ(t): ˆ Z t dχ(t) ˆ ˜ )χ(t ˆ , (21) =− dτ k(τ ˆ − τ ) + R(t) dt 0 where the rate kernel is defined as (22)
ˆ ˆ ˜ ) = (eiQˆ Lt k(τ iLχ, ˆ iLˆχ)( ˆ χ, ˆ χ) ˆ −1 .
ˆ The random reactive flux R(t) is (23)
ˆ ˆ ˆ = eiQˆ Lt R(t) iLχ ˆ.
ˆ Since Trˆ ρ(0)R(t) = 0, the average of Eq. (21) over the initial nonequilibrium ensemble yields the generalized chemical rate law, Z t dχ(t) ¯ ˜ )χ(t (24) =− dτ k(τ ¯ − τ) . dt 0
CHEMICAL RATE LAWS AND RATE CONSTANTS
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Using Eq. (15), we also note that this equation can be written as Z t dCχ (t) ˜ )Cχ (t − τ ) , (25) dτ k(τ =− dt 0
where Cχ (t) = (χ(t), ˆ χ)( ˆ χ, ˆ χ) ˆ −1 is the normalized progress variable autocorrelation function. This equation establishes the fact that we may monitor either the decay of the progress variable fluctuations about equilibrium or the decay of nonequilibrium initial states to determine the rate constant. It is convenient to integrate Eq. (24) over time in order to express the time evolution of the progress variable as Z t ˜ 0 )χ(t (26) χ(t) ¯ − χ(0) ¯ =− dt0 K(t ¯ − t0 ) , 0
which involves the new rate kernel Z t ˆ ˜ ) = (eiQˆ Lt ˜ (27) dτ k(τ χ, ˆ iLˆχ)( ˆ χ, ˆ χ) ˆ −1 . K(t) = 0
This kernel may also be written as
(28)
˜ K(t) =
1 ˆˆ Tr[χ, ˆ eiQLt χ]ˆ ˆ ρe (χ, ˆ χ) ˆ −1 . −i¯ hβ
We next consider the conditions under which the generalized rate laws, Eqs. (24) or (26), reduce to their phenomenological forms, Eqs. (7) or (10), respectively. In order to do this we need to examine the structure of the rate kernels. . 2 4. Structure of the rate kernel. – Since [χ, ˆ χ] ˆ = 0, we see from Eq. (28) that ˜ the initial value of the rate kernel is zero, K(0) = 0. The projection operator Pˆ was constructed to project out of the dynamics of any operator that part which is proportional to the progress variable χ. ˆ Consequently, since the reactive flux autocorrelation function ˜ evolves by projected dynamics iQ ˆ L, ˆ it will decay to zero on a microscopic time scale k(t) provided there are no other slowly varying degrees of freedom in the system other than ˜ χ(t). ˆ (3 ) In this circumstance χ(t) ¯ will decay much more slowly than k(t) and we may take χ(t) ˆ out of the integral in Eq. (24) for times t >> tmic , the microscopic decay time ˜ of k(t). If t∗ is a time such that tmic
